In what follows, the computational complexity of the proposed algorithm is briefly discussed. Unlike conventional massive device detection algorithms, the proposed algorithm does not recover the original unknown signal in the high-dimensional space \(\mathbb {C}^{N\times M}\) . It recovers the sparse signal in a potentially low-dimensional space \(\mathbb {C}^{N\times r^e}\) . Specifically, the computational complexity of the FAT-DL algorithm mainly comes from the matrix multiplication \(\mathbf {AS}\) . Given \(\mathbf {A}\in \mathbb {C}^{L\times N}\)  and \(\mathbf {S}\in \mathbb {C}^{N\times r^e}\) , the computational complexity of FAT-DL is in the order of \(\mathcal {O}(LNr^e)\)  per layer. In contrast with the AMP-based device detection algorithm [1]}, [2]}, the computational complexity of FAT-DL does not grow by increasing the number of BS antennas \(M\) .
FAT-DL learns \(\Omega  \in \mathbb {C}^{N\times 2J}\)  and \(\beta  \in \mathbb {C}^{N\times 1}\)  for inner and outer layers. Thus, its memory complexity can be approximated as \(T_{\max }(N+2NJ\tau _{\max })\)  over \(T_{\max }\)  layers.
